Baseline Predictor

Recommender Systems

Hard

Problem

Theory

A baseline predictor in recommender systems estimates a user's rating for an item using only global statistics and user/item biases, without considering any similarity-based or latent factor information.

The baseline captures the expected rating based on:

• The overall average rating in the system
• How much this user tends to rate above or below average
• How much this item tends to be rated above or below average

The Baseline Formula

b_{ui} = \mu + b_u + b_i

where:

• b_{ui} is the baseline estimate for user u rating item i
• \mu is the global mean rating across all ratings
• b_u is the user bias (user u's deviation from global mean)
• b_i is the item bias (item i's deviation from global mean)

Understanding Each Component

Global mean \mu:

The average of all ratings in the dataset.

\mu = \frac{1}{|R|} \sum_{(u,i) \in R} r_{ui}

User bias b_u:

How much user u tends to rate above or below the global average. A user who rates generously has positive b_u; a harsh critic has negative b_u.

Item bias b_i:

How much item i tends to be rated above or below average. Popular, high-quality items have positive b_i; disliked items have negative b_i.

Computing User and Item Biases

Simple approach (mean deviation):

b_u = \frac{1}{|I_u|} \sum_{i \in I_u} (r_{ui} - \mu)

b_i = \frac{1}{|U_i|} \sum_{u \in U_i} (r_{ui} - \mu)

where I_u is items rated by user u, and U_i is users who rated item i.

Problem: This double-counts the bias. If a generous user rates a popular item, both get credit for the high rating.

Iterative Bias Estimation

A better approach alternates between estimating user and item biases:

Initialize: b_u = 0 for all users, b_i = 0 for all items

Iterate until convergence:

1. Update item biases:

b_i = \frac{\sum_{u \in U_i} (r_{ui} - \mu - b_u)}{|U_i|}

2. Update user biases:

b_u = \frac{\sum_{i \in I_u} (r_{ui} - \mu - b_i)}{|I_u|}

This separates the effects of user generosity from item quality.

Regularized Bias Estimation

To prevent overfitting, especially for users or items with few ratings, add regularization:

b_i = \frac{\sum_{u \in U_i} (r_{ui} - \mu - b_u)}{\lambda_1 + |U_i|}

b_u = \frac{\sum_{i \in I_u} (r_{ui} - \mu - b_i)}{\lambda_2 + |I_u|}

The regularization terms \lambda_1 and \lambda_2 shrink biases toward zero when there is little data.

Worked Example

Ratings:

• User A rates Item 1: 5
• User A rates Item 2: 4
• User B rates Item 1: 3
• User B rates Item 2: 2
• User C rates Item 1: 4

Step 1: Global mean

\mu = \frac{5 + 4 + 3 + 2 + 4}{5} = \frac{18}{5} = 3.6

Step 2: Simple bias calculation

User A: Items rated = {1, 2}, ratings = {5, 4}

b_A = \frac{(5-3.6) + (4-3.6)}{2} = \frac{1.4 + 0.4}{2} = 0.9

User B: Items rated = {1, 2}, ratings = {3, 2}

b_B = \frac{(3-3.6) + (2-3.6)}{2} = \frac{-0.6 + (-1.6)}{2} = -1.1

User C: Items rated = {1}, rating = {4}

b_C = \frac{4-3.6}{1} = 0.4

Item 1: Users = {A, B, C}, ratings = {5, 3, 4}

b_1 = \frac{(5-3.6) + (3-3.6) + (4-3.6)}{3} = \frac{1.4 - 0.6 + 0.4}{3} = 0.4

Item 2: Users = {A, B}, ratings = {4, 2}

b_2 = \frac{(4-3.6) + (2-3.6)}{2} = \frac{0.4 - 1.6}{2} = -0.6

Step 3: Baseline prediction

Predict User C's rating for Item 2:

b_{C,2} = \mu + b_C + b_2 = 3.6 + 0.4 + (-0.6) = 3.4

Why Baselines Matter

They capture a lot of variance:

In many datasets, baselines alone explain 50-70% of rating variance. Users and items have strong inherent biases.

Foundation for other models:

More sophisticated models often predict the residual r_{ui} - b_{ui} rather than the raw rating. This makes the remaining signal easier to learn.

Simple and fast:

Baselines require only counting and averaging. They can be computed efficiently even for large datasets.

Baseline in Matrix Factorization

Matrix factorization models often incorporate baselines:

\hat{r}_{ui} = \mu + b_u + b_i + \mathbf{p}_u^T \mathbf{q}_i

The latent factors \mathbf{p}_u and \mathbf{q}_i model the residual after accounting for biases.

During SGD training, biases are updated alongside latent factors:

b_u \leftarrow b_u + \eta (e_{ui} - \lambda b_u)

b_i \leftarrow b_i + \eta (e_{ui} - \lambda b_i)

where e_{ui} = r_{ui} - \hat{r}_{ui} is the prediction error.

Baseline in Neighborhood Methods

Neighborhood-based collaborative filtering also uses baselines:

\hat{r}_{ui} = b_{ui} + \frac{\sum_{j \in N(i;u)} s_{ij} (r_{uj} - b_{uj})}{\sum_{j \in N(i;u)} |s_{ij}|}

This predicts the baseline plus a weighted average of how neighbors deviate from their baselines.

Time-Aware Baselines

User and item biases can change over time:

User bias drift:

b_u(t) = b_u + \alpha_u \cdot \text{sign}(t - t_u) \cdot |t - t_u|^\beta

where t_u is the user's mean rating time.

Item bias drift:

New items may have inflated ratings initially, then stabilize.

Temporal binning:

Compute separate biases for different time periods.

Handling New Users and Items

New user (no ratings):

Set b_u = 0. Prediction becomes \mu + b_i.

New item (no ratings):

Set b_i = 0. Prediction becomes \mu + b_u.

Cold start:

Both biases are zero. Prediction is just \mu.

With regularization, biases naturally shrink to zero when data is sparse.

Clipping Predictions

Baseline predictions can fall outside the valid rating range:

If ratings are 1-5 and b_{ui} = 5.3, clip to 5.

\hat{r}_{ui} = \max(r_{min}, \min(r_{max}, b_{ui}))

This ensures predictions are always in the valid range.

Baseline as Benchmark

Any recommender system should beat the baseline predictor. If your sophisticated model performs worse than \mu + b_u + b_i, something is wrong.

Typical RMSE improvements:

• Global mean only: RMSE baseline
• With user/item biases: 10-20% improvement
• With collaborative filtering: additional 5-15% improvement

Decomposing Ratings

The baseline model decomposes each rating:

r_{ui} = \mu + b_u + b_i + \text{interaction} + \text{noise}

• \mu: What is the typical rating?
• b_u: Is this user generous or harsh?
• b_i: Is this item generally liked or disliked?
• interaction: Does this specific user like this specific item?
• noise: Random variation

The baseline captures the first three components.

Examples

Hints

1. Build user means from rows and item means from columns while skipping zeros.
2. For each pair, add the global mean, its user bias, and its item bias.

Requirements

• Compute the global mean from nonzero ratings only.
• Compute user and item biases from their observed ratings only.
• Use zero bias when a user or item has no observed ratings.
• Return one prediction for each pair in target_pairs.

Constraints

• ratings_matrix is rectangular and contains at least one nonzero rating.
• A zero entry represents a missing rating.
• Every target pair contains valid user and item indices.
• Time limit: 300 ms.

Starter Code

def baseline_predict(ratings_matrix: list, target_pairs: list) -> list:
    """
    Returns the baseline predictions for the requested user-item pairs.
    """
    # Write code here
    pass

Test Cases